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The P k Partition Problem and Related Problems in Bipartite Graphs

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SOFSEM 2007: Theory and Practice of Computer Science (SOFSEM 2007)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4362))

Abstract

In this paper, we continue the investigation proposed in [15] about the approximability of P k partition problems, but focusing here on their complexity. More precisely, we prove that the problem consisting of deciding if a graph of nk vertices has n vertex disjoint simple paths {P 1, ⋯ ,P n } such that each path P i has k vertices is NP-complete, even in bipartite graphs of maximum degree 3. Note that this result also holds when each path P i is chordless in G[V(P i )]. Then, we prove that the optimization version of these problems, denoted by Max P 3 Packing and MaxInduced P 3 Packing, are not in PTAS in bipartite graphs of maximum degree 3. Finally, we propose a 3/2-approximation for Min3-PathPartition in general graphs within O(nm + n 2logn) time and a 1/3 (resp., 1/2)-approximation for MaxW P 3 Packing in general (resp., bipartite) graphs of maximum degree 3 within O(α(n,3n/2)n) (resp., O(n 2logn)) time, where α is the inverse Ackerman’s function and n = |V|, m = |E|.

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References

  1. Arkin, E., Hassin, R.: On Local Search for Weighted Packing Problems. Mathematics of Operations Research 23, 640–648 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  2. Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., Protasi, M.: Complexity and Approximation (Combinatorial Optimization Problems and Their Approximability Properties). Springer, Heidelberg (1999)

    MATH  Google Scholar 

  3. Bazgan, C., Hassin, R., Monnot, J.: Approximation Algorithms for Some Routing Problems. Discrete Applied Mathematics 146, 3–26 (2005)

    Article  MathSciNet  Google Scholar 

  4. Chazelle, B.: A Minimum Spanning Tree Algorithm with Inverse-Ackermann Type Complexity. J. ACM 47, 1028–1047 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  5. Csaba, B., Karpinski, M., Krysta, P.: Approximability of Dense and Sparse Instances of Minimum 2-Connectivity, TSP and Path Problems. In: SODA, pp. 74–83 (2002)

    Google Scholar 

  6. Dyer, M., Frieze, A.: Planar 3DM is NP-Complete. J. Algorithms 7, 174–184 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  7. Frank, A.: Some Polynomial Time Algorithms for Certain Graphs and Hypergraphs. In: Proceedings of the 5th British Combinatorial Conference. Congressus Numerantium, vol. XV, pp. 211–226. Utilitas Mathematicae, Winnipeg (1976)

    Google Scholar 

  8. Garey, M.R., Johnson, D.S.: Computers and Intractability. A guide to the Theory of NP-Completeness. W.H. Freeman, New York (1979)

    MATH  Google Scholar 

  9. Hassin, R., Rubinstein, S.: An Approximation Algorithm for Maximum Packing of 3-Edge Paths. Inf. Process. Lett. 63, 63–67 (1997)

    Article  MathSciNet  Google Scholar 

  10. Hassin, R., Rubinstein, S.: An Approximation Algorithm for Maximum Triangle Packing. In: Albers, S., Radzik, T. (eds.) ESA 2004. LNCS, vol. 3221, pp. 403–413. Springer, Heidelberg (2004)

    Google Scholar 

  11. Kaneko, A.: A Necessary and Sufficient Condition for the Existence of a Path Factor Every Component of Which is a Path of Length at Least Two. Journal of Combinatorial Theory 88(B), 195–218 (2003)

    Article  MATH  Google Scholar 

  12. Karpinski, M.: Personnal communication (2006)

    Google Scholar 

  13. Kirkpatrick, D.G., Hell, P.: On the Completeness of a Generalized Matching Problem. In: Proc. STOC’78, pp. 240–245 (1978)

    Google Scholar 

  14. Lovasz, L., Plummer, M.D.: Matching Theory. North-Holland, Amsterdam (1986)

    MATH  Google Scholar 

  15. Monnot, J., Toulouse, S.: Approximation Results for the Weighted P4 Partition Problem. In: Liśkiewicz, M., Reischuk, R. (eds.) FCT 2005. LNCS, vol. 3623, pp. 377–385. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  16. Monnot, J., Toulouse, S.: The P k Partition Problem and Related Problems in Bipartite Graphs. Technical Report (2006), available at http://www.lamsade.dauphine.fr/ monnot/publications(journal).htm

    Google Scholar 

  17. Petrank, E.: The Hardness of Approximation: Gap Location. Computational Complexity 4, 133–157 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  18. Steiner, G.: On the k-Path Partition Problem in Cographs. Cong. Numer. 147, 89–96 (2000)

    MATH  MathSciNet  Google Scholar 

  19. Steiner, G.: On the k-Path Partition of Graphs. Theor. Comput. Sci. 290, 2147–2155 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  20. Wang, H.: Path Factors of Bipartite Graphs. Journal of Graph Theory 18, 161–167 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  21. Vishwanathan, S.: An Approximation Algorithm for the Asymmetric Travelling Salesman Problem with Distances One and Two. Information Processing Letter 44(6), 297–302 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  22. Yan, J.-H., Chang, G.J., Hedetniemi, S.M., Hedetniemi, S.T.: k-Path Partitions in Trees. Discrete Applied Mathematics 78, 227–233 (1997)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. CNRS LAMSADE - UMR 7024, Université Paris-Dauphine, Place du Maréchal De Lattre de Tassigny, 75775 Paris Cedex 16, France

    Jérôme Monnot

  2. LIPN - UMR CNRS 7030, Institut Galilée, Université Paris 13, 99 av. Jean-Baptiste Clément, 93430 Villetaneuse, France

    Sophie Toulouse

Authors
  1. Jérôme Monnot
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  2. Sophie Toulouse
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Editor information

Jan van Leeuwen Giuseppe F. Italiano Wiebe van der Hoek Christoph Meinel Harald Sack František Plášil

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Monnot, J., Toulouse, S. (2007). The P k Partition Problem and Related Problems in Bipartite Graphs. In: van Leeuwen, J., Italiano, G.F., van der Hoek, W., Meinel, C., Sack, H., Plášil, F. (eds) SOFSEM 2007: Theory and Practice of Computer Science. SOFSEM 2007. Lecture Notes in Computer Science, vol 4362. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69507-3_36

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  • DOI: https://doi.org/10.1007/978-3-540-69507-3_36

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-69506-6

  • Online ISBN: 978-3-540-69507-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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